When \(\mu _1>\mu _0\)  the distribution of \(\pi _{t,1}^{\text{TS}}\)  is skewed towards 1, and converges to 1 as \(t \rightarrow \infty \)  [1]}. This property explains the large imbalance between arms' sample sizes, generally favouring arm 1. When \(\mu _1=\mu _0\) , the distribution of \(\pi _{t,1}^{\text{TS}}\)  does not concentrate [2]} and may result (though less likely) in extremely imbalanced allocations as well, favouring either arm 1 or 0. Supplementary material A4 provides illustrative examples.
Such highly exploitative behaviour of TS negatively impacts the theoretical asymptotic distribution of traditional statistical estimators, motivating the need for specific TS-collected data techniques.
